We propose a fully conformal invariant theory describing gravity as a spontaneously broken theory. Newton's constant is automatically generated. We tind through the study of classical solutions of the equations of motion that the breakdown of conformal symmetry can take place at the tree approximation without introducing arbitrary forms for the scalar potential. Using cosmological metrics, which we find natural from the physical point of view, some conclusions can be drawn regarding the nature of those metrics. The case of constant scalar curvature is particularly interesting, and gives rise to a gravitational version of the Goldstone theorem.
I. INTRODUCTION
Recently, the idea of spontaneously generated gravity has been seriously considered by several authors [I]. The main point is that gravity behaves somewhat like weak interactions. It has a dimensional coupling-Newton's constant-carrying dimensions of an inverse squared mass. This is reminiscent of the old Fermi version of weak interactions, where the Fermi constant G, is also dimensional as well as proportional to an inverse squared mass. In the past, Fermi's theory was considered non-renormalizable due to the presence of this dimensional coupling. However, when unified with electromagnetic interactions in a larger non-abelian gauge theory, G, appears to be related to the inverse of the squared vacuum expectation value of some scalar fields which are responsible for the breakdown of the gauge invariance. This unified theory of electroweak phenomena, widely known as the Standard Model of Glashow-Weinberg-Salam, is indeed renormalizable and it has been exceedingly succesful in describing those interactions in particle physics.
The question is: Would it be possible to describe gravity through a highly symmetric theory with no dimensional couplings? If so, Newton's constant might appear as related to the vacuum expectation value of the fields breaking such a symmetry. Theories with no dimensional couplings are welcomed by particle physicists and they are probably the best candidates for a full unification of all interactions in nature.
So far, the only lagrangian (with very few changes) proposed to carry out this program is scale (in fact conformal) invariant, but the breakdown of the symmetry is, 64